# Find the equation of the plane parallel to two lines

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1. Sep 6, 2016

### Sho Kano

1. The problem statement, all variables and given/known data
Let A, B and C be distinct vectors in V3 with B and C non-parallel.
a. Find an equation for the plane containing both the line through A parallel to B and the line through A parallel to C.
b. Verify that the two lines actually lie in the plane.

2. Relevant equations

3. The attempt at a solution
I can't see any situation where a plane containing A can be parallel to a plane containing C if they are distinct vectors (distinct as in non-parallel)

2. Sep 6, 2016

### Staff: Mentor

What is V3? Presumably it's a 3-dimensional vector space. If so, is it different from $\mathbb{R}^3$?
This is a very strange description. The context here seems to be that A is a point, but the earlier description is that A is a vector. I would guess that what they're calling vector A is the vector $\vec{OA}$, with O being the origin and A being the endpoint of the vector. Otherwise this problem doesn't make much sense.

Was the problem statement originally in some other language?

3. Sep 6, 2016

### andrewkirk

The question is not well expressed. Here is what I think they mean.

Let A,B,C be three points in $\mathbb R^3$ and denote the origin by $O$.

Let L1 be a line through A that is parallel to line segment $\bar{OB}$.
Let L2 be a line through A that is parallel to line segment $\bar{OC}$.

(b) Show that there is a plane that contains both L1 and L2; and
(a) find the equation of that plane.

4. Sep 6, 2016

### Ray Vickson

While the question is not absolutely clear, I think one interpretation of it is that we have two vectors $\vec{B}$ and $\vec{C}$ in a 3-dimensional space; these are, perhaps, like force vectors or velocity vectors, having tails and heads not necessarily at the origin. Assuming that is the case, make a copy $\vec{C}'$ of $\vec{C}$ but whose tail coincides with the tail of $\vec{B}$, so that $\vec{B}$ and $\vec{C}'$ emanate from the same point in space, but have different directions. There certainly IS a plane P that contains both vectors $\vec{B}$ and $\vec{C}'$ (that is, which contains the three points that lie at the two ends of these vectors). Any Plane P that is parallel to P will be parallel to both vectors $\vec{B}$ and $\vec{C}$ (the original $\vec{C}$), and now all you need to is figure out which such plane passes through the point A which lies at the end of the vector $\vec{A}$ (assuming that this last vector has its tail at the origin).

5. Sep 7, 2016

### Sho Kano

I copied the problem straight off the problem set; I'll try to clarify with the teacher today

6. Sep 7, 2016

### Sho Kano

There's no difference, my teacher just likes to use V3 as a distinction from R3 (for whatever reason) for the first few days of the class

7. Sep 7, 2016

### Sho Kano

OK so apparently, we are supposed to find a plane that has both B and C in it, and A intersecting it. It makes sense because a plane can be parallel to many vectors.
The equation of such a plane can be $\left( \left< x,\quad y,\quad z \right> -\overrightarrow { OA } \right) \cdot \overrightarrow { B } \times \overrightarrow { C } =0$ right?
Then I guess I just do part b by plugging in OA + tB and OA + tC.

Last edited: Sep 7, 2016